Singular Soliton Operators and Indefinite Metrics
Abstract
The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All algebro-geometrical or "singular finite-gap" potentials satisfy to this condition. A Spectral Theory is constructed here for the periodic and rapidly decreasing cases in the special classes of functions with singularities and indefinite inner product. It has a finite number of negative squares if the unimodular Bloch multipliers are fixed in the periodic case, and in the rapidly decreasing case. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform for the Riemann Surfaces preserving remarkable multiplicative properties of the ordinary (i.e. genus zero) Fourier Transform based on the standard exponential basis, leads to such operators as it was shown in our previous works.
Cite
@article{arxiv.1103.2505,
title = {Singular Soliton Operators and Indefinite Metrics},
author = {P. G. Grinevich and S. P. Novikov},
journal= {arXiv preprint arXiv:1103.2505},
year = {2015}
}
Comments
LaTex, 34 pages, 3 figures. Some references were corrected. Additional results about decomposition by eigenfunctions of complex singular finite-gap potentials, higher order potentials and non-statiobnary Schrodinger operator were added